1. Trigonometric Graphs
1.6
Day 1

2. Let’s first make a chart of this function:
y = sin x x 30 60 90
120
150
180
210
240
270
300
330
360
sin x
0.5
0.87 1
0.87
0.5
-0.5
-0.87 -1
-0.87
-0.5

3. Now let’s plot 1
0.5
-0.5 -1

4. y = sinx Maximum Value = 1 Domain: All Reals Range: -1 ≤ y ≤ 190
180
270
360 -1
Maximum Value = 1
Domain: All Reals
Range: -1 ≤ y ≤ 1
Minimum Value = -1

5. Let’s first make a chart of this function:
y = cos x x 30 60 90
120
150
180
210
240
270
300
330
360
cos x 1
0.87
0.5
-0.5
-0.87 -1
-0.87
-0.5
0.5
0.87 1

6. y = cosx Maximum Value = 1 Domain: All Reals Range: -1 ≤ y ≤ 190
180
270
360 -1
Maximum Value = 1
Domain: All Reals
Range: -1 ≤ y ≤ 1
Minimum Value = -1

7. Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an x-interval of
6. The cycle repeats itself indefinitely in both directions of the x-axis.
Properties of Sine and Cosine Functions

8. If |a| > 1, the amplitude stretches the graph vertically.
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis. y x
y = 2sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Amplitude

9. The period of a function is the x interval needed for the function to complete one cycle.
For b  0, the period of y = a sin bx is
Shrink Horizontally: y x
period:
period: 2π
Stretch Horizontally: y x
period: 4
period: 2π

10. Y = 7sinx 7 90
180
270
360 -7
Maximum Value = 7
Minimum Value = -7

11. What is the equation of this graph? Maximum Value = 4
Y = 4cosx 4 90
180
270
360 -4
What is the equation of this graph?
Maximum Value = 4
Minimum Value = -4

12. What is the equation of this graph? Maximum Value = 8
Y = - 8sinx 8 90
180
270
360 -8
“Opposite” to Sin x
What is the equation of this graph?
Maximum Value = 8
Minimum Value = -8

13. Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x. y x
Use the identity cos (–x) = – cos x
y = cos (–x)
y = cos (–x)
Graph y = f(-x)

14. Use the identity sin (– x) = – sin x:
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
Use the identity sin (– x) = – sin x:
y = 2 sin (–3x) = –2 sin 3x
period: 2 3 =
amplitude: |a| = |–2| = 2
Calculate the five key points. 2 –2
y = –2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
Example: y = 2 sin(-3x)

15. Graph of the Tangent Function
To graph y = tan x, use the identity
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = tan x
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
period:
Tangent Function

16. Example: Tangent Function
Example: Find the period and asymptotes and sketch the graph of y x
1. Period of y = tan x is .
2. Find consecutive vertical
asymptotes by solving for x:
Vertical asymptotes:
3. Plot several points in
4. Sketch one branch and repeat.
Example: Tangent Function

17. Graph of the Cotangent Function
To graph y = cot x, use the identity
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = cot x
vertical asymptotes
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:

18. Graph of the Secant Function
The graph y = sec x, use the identity
At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y x
Properties of y = sec x
1. domain : all real x
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
Secant Function

19. Graph of the Cosecant Function
To graph y = csc x, use the identity
At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. x y
Properties of y = csc x
1. domain : all real x
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
where sine is zero.
Cosecant Function

20. Y = -9sinx 9 90
180
270
360 -9
“Opposite” to Sin x

21. Y = sin 2x Period of graph is 180090
180
270
360 -1
Period of graph is 1800
There are 2 cycles between 00 and 3600

22. Combining these rules Draw y = 6sin2x Y = 6sin 2x Max 6 2 cycles
Min -6
Period = 360 ÷ 2 = 1800 6
Y = 6sin 2x 90
180
270
360 -6

23. Recognising Graph Y = 8cos4x Max 8 4 cycles Cosine Min -8 8 90 180 27090
180
270
360 -8

24. Combining our two rules Draw y = 8sin2x
Max 8
2 cycles
Min -8
Period = 360 ÷ 2 = 1800 8
Y = 8sin 2x 90
180
270
360 -8